This paper showed that substituting m=4π into the kinetic term of the Schrödinger equation gives −(1/8π)∇²ψ + Vψ = Eψ, and the first edition claimed that this 8π has the same origin as the 8πG of Einstein's equation. The second edition withdrew that claim by separating three mutually unrelated factors of 2. The 3rd edition carries that separation to the end.
First, this 8π does not arise in the series' own system of units. Obtaining ħ²/(2m)=1/(8π) requires two choices, ħ=1 and m=4π. But as the 3rd editions of Papers 38 and 39 established, at most three values may be freely assigned against the base dimensions M, L, T, and the series has spent them on m=4π, G=4π, ħ=1/(4π) (so that c is no longer choosable). In that system the coefficient is ħ²/(2m)=(1/4π)²/(2·4π)=1/(128π³)=0.0002519651, a factor 16π²=157.9137 away from 1/(8π)=0.03978874. The 8π therefore appears only when ħ=1 is adopted, and ħ=1 and ħ=1/(4π) are incompatible: units can be fixed only once. The check of the 3rd edition of Paper 15 here operates inside the series itself — a convention must be a number, and at the same time must not be two different numbers.
Second, the coefficient of the Laplacian is not itself measurable. Applying x→λx to the free-particle equation −A d²ψ/dx² = Eψ scales it as A→A/λ²; as verified here, A=1/(8π)=0.03978874 becomes 0.00994718 at λ=2 and 0.00158314 at λ=√(8π). Its value can be made any positive number purely by the choice of the unit of length, and on its own it contains no physics. What is measurable is a dimensionless ratio: computed here for the ground state of hydrogen, E_1/(m_e c²) = −13.605693 eV / 510999.0 eV = −2.662568×10⁻⁵, agreeing exactly with −α²/2 = −2.662568×10⁻⁵ obtained from α=1/137.035999. That number is the same in every system of units; the 8π is not. The discipline of the 3rd edition of Paper 38 — dimensionless ratios are the physics — applies here too.
Third, each of the three factors of 2 is given its own address. The second edition showed that the three are mutually unrelated; the 3rd edition writes where each belongs. Schrödinger's 2 comes from ∫mv dv = ½mv², i.e. from analysis, and is unrelated to geometry. Einstein's 2 is the trace reversal R_μν − ½R g_μν, which is forced by the contracted Bianchi identity ∇_μ G^μν ≡ 0 — the address of Paper 107 (Noether's second theorem: identities given by local symmetry). Spin's 2 is the double cover SU(2)→SO(3) — the address of Paper 104 (Clifford algebra, 720°) and Paper 68 (the Hopf map S³→S²). That Einstein's 2 is precisely the ½ of the trace reversal can be checked in the weak-field limit: for dust with signature (−,+,+,+), T_00 − ½T g_00 = ½ρc², and since R_00 ≃ ∇²Φ/c² this gives ∇²Φ = ½κρc⁴, which together with Poisson's ∇²Φ = 4πGρ yields ½κc⁴ = 4πG, i.e. κ = 8πG/c⁴. And only once the address is written does "distinct roots" become a checkable claim. Writing merely that the three are unrelated leaves the reader no way to verify it; writing where each belongs lets one go and look at the Bianchi identity, at the Clifford algebra, at the integral, and check each independently. The term "distinct roots" established in Papers 86 and 100 works only once the destination is written.
Fourth, the 4πs are distinct roots too, so nothing at all in the two 8πs coincides. On Einstein's side: 4π (Gauss's flux, i.e. solid angle — forced) × 2 (trace reversal, Bianchi — forced). On Schrödinger's side: 4π (the unit m=4π — chosen) × 2 (the integral of ½mv² — analysis). Since the roots differ factor by factor, the two 8πs share only a numerical value. The second edition wrote that "what they share is the underlying 4π"; the 3rd edition withdraws this too. Just as the 3rd edition of Paper 35 sorted the two 4πs inside 4πG into a fact and a convention, here too it is not "the same 4π": one cannot be chosen, the other was chosen.
Fifth, agreement with Paper 18 is no cause for surprise. For a plane wave ψ=e^(ikx) with k=4πv, ⟨E_k⟩ = k²/(8π) = (4πv)²/(8π) = 2πv², agreeing with E_k = 2πv² of Paper 18 (checked numerically at v=1, 2, 0.5). But both come from the same substitution m=4π. In the census of the 3rd edition of Paper 16 this is root (b), a consequence of substitution, and it says nothing about the world. It is not a "transfer of machinery" in the sense of Paper 100.
Finally, two things left standing in the second edition have been removed. (i) The end of the second edition's abstract still carried wording from the first edition — "clarifying the double structure of the 8π guardrail (the meat lid) and the 4πr² surface area (the skin data), and achieving perfect synchronisation …" — but Section 4 of the body had already withdrawn exactly that expression. The abstract and the body disagreed. The same accident occurred in the second edition of Paper 35, where only the English abstract had been updated, leaving a withdrawn claim in the Japanese one. At every revision, check the abstract against the body first. (ii) The naming of 8π in the theorem ("the universal rendering-cost coefficient governing the curvature dissipation of waves in three-dimensional space / 8π guardrail / meat lid") has also been removed from the body, since Paper 82 quarantines this kind of language (rendering, cost, infrastructure) in Layer 3, the non-scientific layer. Naming is not deriving: this paper does not show that 8π "governs curvature dissipation", and the value itself is an artefact of the units. All numbers were computed here (no external tables consulted). This paper does not modify quantum mechanics and claims no isomorphism of the two 8πs.
(scope note) Positioning of this paper: This paper offers a structural interpretation of established physical laws and mathematical theorems. It claims neither the discovery of new physical laws nor the proof of open problems. The algebraic relations presented are transformations of known expressions; the aim is to propose a geometric reading of them. No claim contradicting established theory is made.
A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. The aim is a modest one: to make a few recurring geometric structures — in particular the solid angle 4π and the exponent n = 2 — easier to see within a single picture. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. Nothing here is intended to contradict existing results.
On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The $8\pi$ Is an Artefact of the Units [Paper 17]. Zenodo.
DOI: 10.5281/zenodo.21600813
ORCID: 0009-0006-8587-3450